Lesson video

Hello, I'm Mr. Coward, and welcome to today's lesson on solving adfected quadratic equations, part three.

For today's lesson, all you'll need is a pen and paper, or something to write on and with.

If you can, please take a moment to clear away any distractions, including turning off any notifications.

And if you can, try and find a quiet space to work where you won't be disturbed.

Okay, when you're ready, let's begin.

Okay, so time to try this task.

Create three different quadratic equations with solutions x equals 11 and x equals negative 5.

So I have a hint, but I'd like you to pause the video first.

So pause the video and have a go.

Pause in three, two, one.

Okay, here's my hint.

So say I do it with different numbers.

Say I had x equals negative 7 and x equals positive 2.

Okay, so they were my solutions.

I'm going to work backwards here.

So we'd have x minus 7, oh, sorry, x add 7, add in 7 to both sides, equals 0, or, and I'm not 100% sure why I put that in brackets there.

x, subtract 2 from both sides.

x minus 2 equals 0.

So now is when we can put in brackets.

Now we can say that this equals 0.

But, do you remember from a couple of lessons ago, when this, when we did this, when something like that had the same solution as something like this.

15x minus 30, or 3x minus 6, or 10x minus 20, or 20x minus 40, okay? So, pause the video now and have a go, if you haven't already.

Okay, so welcome back, so here are my possible answers.

You'd say that x is equal to 11 or x is negative 5.

So that means if we rearrange this, we get this, equals 0, and now we can say that this equals 0 or this equals 0, so we can have this equals 0, so just working backwards, and then you might want to choose to do, so, we've got a quadratic equation, x squared minus, so this could be one of them.

Well, this is technically one of them, but it's just written in a different form.

6x minus 55 equals 0, so that's one of them, or you could have said this.

You could have times both terms in that first bracket by 2, and then expanded that, and then that would give you another one, or you could have times both terms in this bracket by 3, and expanded that, and that would give you another one, or, you could have took this one, and done this.

Add in 1 both sides.

You could have done that.

And we're going to look at this.

We're going to look at stuff like this today's lesson.

So, we're going to create three quadratic equations with the same solution as this.

So what I'm going to do first is I'm going to write my quadratic here.

Now, let's say I add on 20 to both sides.

That will still, because I've done the same thing to both sides, that will still give me the same solution.

So this has the same answers as this.

So that's one possible way you could have done it.

Okay, what about now, if I don't want to, what about now, but change this.

What if I take 7x from both sides? Okay, that's different.

It'll give me the same solutions.

What about now if I, you know what, let's get really complicated.

What if I add on 3x squared and add on 10 to both sides? So, that gives me 4x squared plus 7x minus 10, no, 8 now, yeah, 8, minus 8 equals 3x squared plus 10.

Okay, that looks a lot more complicated, but it'll give us the same solutions.

So what we can do is we can change our equation.

We can do the same thing to both sides and our solutions are maintained.

So now, we're going to use this principle that we could do the same thing to both sides and our solutions are maintained to solve the following quadratic equations.

So, do you remember when I said that we want our quadratic equations to be equal to 0? That means that it's easier because then we can factorise and say one bracket equals 0 or the other bracket equals 0.

That is the best way to solve, well, not the best way, but that is how we are going to solve quadratic equations.

There's other ways, but we're not going to learn about them yet.

So, we need to rearrange this to equal 0, and if I rearrange it, do my solutions change? No, not if I do the same thing to both sides.

So how can I get this equals to 0? Well, I can get rid of this 8x.

That makes that equal to 0.

Now what must I do on this side? Well, I must also take away 8x, and I'm going to take with x, I'm going to write that in the middle just because I like to have it in, in the order of my highest power going to my lowest power, yeah.

So, I've rearranged that.

Now I factorise.

Okay, so now x equals 6 or x equals 2.

Yeah, sorry, yeah, I've skipped a step.

x this bracket equals 0 or this bracket equals 0.

So x equals 6 or x equals 2.

So they are my two solutions.

So, rearrange to get equal to 0, factorise, one bracket equals 0 or the other bracket equals 0, and then solve those two equations separately.

Okay, have a go at this one.

So, pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Now, hopefully, you've rearranged it successfully, taking 15 from both sides.

Now we factorise.

So they have a different side and they add to negative 2.

So x minus 5 equals 0 or x plus 3 equals 0.

I didn't write in my or sign there.

So x equals 5 or x equals negative 3.

So there are our two solutions.

Really well done if you got them.

Okay, that's not going to work, is it? All right, let's rearrange this.

So, what do we do? Well, we want to get one side equal to 0.

Now I could subtract the x squared and get that side equal to 0.

That would work.

However, I don't really like to factorise negative x squareds.

I prefer to factorise with positive x squareds.

So because of that, I'm going to take these onto the other side.

Okay, I'm doing that because I do not like to factorise negative x squareds.

You can, but I don't like it.

I'm going to write them in that order.

I just slightly prefer to write, have my powers from squared to power 1, and actually, this is to the power 0 if you think about it, because x to the power 0 is 1.

Okay, so here we are.

Okay, so here we are.

We factorise that as before.

Either the first bracket is equal to 0 or the second one is, and then we solve that.

Okay, so have a go at this one.

Pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Hopefully, you've rearranged it correctly.

Now, why did I get them on this side? Well, because I don't like to have a negative x squared.

So I took them over to that side, and I factorised that, so that equals 0 or that equals 0.

So x equals 15 or x equals 1.

Okay, so they are my two different solutions there.

Okay, solve the following.

Well, this looks really, really complicated, doesn't it? But we do the same thing.

We rearrange one side to equal 0.

Doing the same thing to both sides.

Doing the same thing to both sides.

So now, we get 1x squared.

Oh, we actually don't need to write our 1, do we? Plus 7x.

That's just equal to 0, and now we factorise.

So we get x plus 6 equals 0 or x plus 1 equals 0.

So we get x equals negative 6 or x equals negative 1.

Okay, not too bad, right? So, your turn.

Pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

So hopefully, you've rearranged it.

Oh, my 6 goes there.

Remember, I said it's easier if you write them underneath.

It just, it's just a bit easier to see, like column addition.

Okay, that's equal to 0, that's equal to 0.

Now you factorise that.

So one bracket is equal to 0 or the next bracket is equal to 0.

So then there, oops, 7, there are my two solutions, x equals negative 3 or x equals negative 7.

So the only extra step is we just have to make one side equal to 0, and a few lessons ago, we did lots of rearranging, so hopefully you'll be a whiz at that.

Okay, what about this one? This one looks really complicated, but it's actually not so bad.

So what I'm going to do is I'm going to expand this side.

Okay, and now.

And now we just do what we did before, we just rearrange it.

1x squared.

We don't need to write the 1.

We keep that secret.

Okay, so x squared minus 10x plus 25, so that gives me that when I factorise it.

So we've just got one solution, x equals 5, okay? 'Cause x minus 5 equals 0, so that means x must be equal to 5.

Okay, so your turn.

So pause the video and have a go at that.

Okay, so hopefully, you've had a go, and hopefully, you've got this.

Okay, take 10 from both sides, and take 2x from both sides.

Now we factorise that.

Two things that times to get negative 20 and add to get positive 8.

So we get x equals negative 10 or x equals 2 because that equals 0 or that equals 0.

So x plus 10 equals 0 or x plus 2 equals 0.

Okay, so know just this with my example from the start of last lesson, one of the ones that I got you to substitute things into.

And I said we'd learn how to do it this lesson.

So, hmm, we can expand this.

Now we can rearrange it.

Oops, sorry, I'll simplify first.

Now we can rearrange it.

And now, we can factorise and solve.

So, expand, rearrange, factorise, and solve.

Right, slung it right in here, isn't it? Add in 5 to both sides, taken 7 from both sides.

Okay, so now your turn.

I would like you to pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Hopefully, you expanded this correctly and got this as your answer.

Now, hopefully, you've rearranged by adding 1 to both sides.

And now factorise.

That's a 4.

And now solve.

Taken 4 from both sides.

So really well done if you got that correct.

Okay, so, final example.

Now, why is this a quadratic? Well, because when I multiply both sides by x, I will get an x squared, and don't forget to multiply the 10 by x.

We're multiplying everything here by x, the whole of that side, so we should get that.

Okay, that and that simplify to 1.

Now we rearrange.

So we get this.

Okay, and now we solve.

Oh, sorry, so that will be x plus 5 and x plus 5 so x plus 5 squared.

Take 5, take 5.

I didn't need the brackets there.

So x equals negative 5.

Okay, so your turn, I'd like to pause video and have a go.

Pause in three, two, one.

Okay, welcome back.

Hopefully, you multiplied both sides by x.

I've got this.

Oh, equals 10, no, plus 10 equals 10, sorry.

Rearrange to get that.

So that means x equals.

So x plus 5 equals 0 or.

Adding 2 to both sides, taking 5 from both sides.

Okay, so really well done if you've got that.

Awesome, so now it's time for the independent task.

So I would like you to pause the video to complete your task and resume once you've finished.

Okay, welcome back.

Here are my answers.

Okay, so now it's time for the explore task.

I would like you to solve the following quadratic equations.

What do you notice? Can you find another set of equations like this? Can you explain it? Can you generalise? So pause the video to complete your task and resume once you're finished.

Okay, welcome back.

Now here are my solutions.

What do you notice? Well, this is going up by 1.

And this is going down by 1.

What's going on with these numbers? What's special about these numbers? 0 squared, 1 squared, 2 squared, 3 squared, 4 squared, 5 squared, 6 squared, 7 squared.

They're our square numbers.

Why? Changing them like that.

Does that mean that this happens? Did you manage to find another set of equations like that? Why does this happen? Well, when we're factorising that, we're looking for two numbers to times to get a 36, and ad to get 12, okay.

So these times to get 36 and they add to get 12.

Now, if I did 36, so I'll concentrate on this one for now, if I did the 36 take away 1, that's equal to 35.

Well, 35 equals 5 times 7.

Interesting.

Okay, just choose a different number.

Say it wasn't 36.

Now let's say it was, I'm deliberately going to pick a square number.

Let's say it was 49, and I did 49 take away 1.

So 7 times 7, which gives me 48.

Well, 48 is, okay, that's interesting.

All right, let's try a different number.

Let's try 100.

100 is another squared number.

100 take away 1, 99.

99 equals 9 times 11.

That's interesting.

What's going on here? Well, any square number minus 1, can you factorise that using difference of two squares? Oh, so what about if we did it with numbers now? So if we had 100, so 100, which is 10 squared, so 10 squared minus 1, which would factorise to 10 plus 1, 10 minus 1.

So that's our 11, and that's our 9.

We're using difference of two squares here.

Okay, so what about this one? Can you now explain it for that? Can you explain it for that? Can you explain it for that? Okay, I've given you a start and I won't tell you too much more, so if you feel like you've got some ideas now, it's a good opportunity for you to continue to explore this task.

Okay, so that is all for this lesson.

Thank you very much for your hard work.

I hope you enjoyed it and found it interesting, and I look forward to seeing you next time.

Thank you.